Post-Newtonian expansion

Expansion in h

Another approach is to expand the equations of general relativity in a power series in the deviation of the metric from its value in the absence of gravity.

${\displaystyle h_{\alpha \beta }=g_{\alpha \beta }-{\eta }_{\alpha \beta }.}$

To this end, one must choose a coordinate system in which the eigenvalues of ${\displaystyle h_{\alpha \beta }{\eta }^{\beta \gamma }}$ all have absolute values less than 1.

For example, if one goes one step beyond linearized gravity to get the expansion to the second order in h:

${\displaystyle g^{\mu \nu }\approx {\eta }^{\mu \nu }-{\eta }^{\mu \alpha }{h}_{\alpha \beta }{\eta }^{\beta \nu }+{\eta }^{\mu \alpha }{h}_{\alpha \beta }{\eta }^{\beta \gamma }{h}_{\gamma \delta }{\eta }^{\delta \nu }.}$

${\displaystyle \sqrt{-g}\approx 1+\frac{1}{2}{h}_{\alpha \beta }{\eta }^{\beta \alpha }+\frac{1}{8}{h}_{\alpha \beta }{\eta }^{\beta \alpha }{h}_{\gamma \delta }{\eta }^{\delta \gamma }-\frac{1}{4}{h}_{\alpha \beta }{\eta }^{\beta \gamma }{h}_{\gamma \delta }{\eta }^{\delta \alpha }.}$

Expansions based only on the metric, independently from the speed, are called post-Minkowskian expansions (PM expansions).

Uses

The first use of a PN expansion (to first order) was made by Albert Einstein in calculating the perihelion precession of Mercury's orbit. Today, Einstein's calculation is recognized as a common example of applications of PN expansions, solving the general relativistic two-body problem, which includes the emission of gravitational waves.

Newtonian gauge

In general, the perturbed metric can be written as

${\displaystyle d{s}^2=a^2(\tau )\left[(1+2A)d{\tau }^2-2B_{i}d{x}^{i}d\tau -\left({\delta }_{ij}+h_{ij}\right)d{x}^{i}d{x}^j\right]}$

where ${\displaystyle A}$, ${\displaystyle B_i}$ and ${\displaystyle h_{ij}}$ are functions of space and time. ${\displaystyle h_{ij}}$ can be decomposed as

${\displaystyle h_{ij}=2C{\delta }_{ij}+{\mathrm{\partial }}_i{\mathrm{\partial }}_{j}E-\frac{1}{3}{\delta }_{ij}{◻}^{2}E+{\mathrm{\partial }}_i{\hat{E}}_j+{\mathrm{\partial }}_j{\hat{E}}_i+2{\tilde{E}}_{ij}}$

where ${\displaystyle ◻}$ is the d'Alembert operator, ${\displaystyle E}$ is a scalar, ${\displaystyle {\hat{E}}_i}$ is a vector and ${\displaystyle {\tilde{E}}_{ij}}$ is a traceless tensor. Then the Bardeen potentials are defined as

${\displaystyle \mathrm{\Psi }\equiv A+H(B-E^{\prime }),+(B+E^{\prime }{)}^{\prime },\mathrm{\Phi }\equiv -C-H(B-E^{\prime })+\frac{1}{3}◻E}$

where ${\displaystyle H}$ is the Hubble constant and a prime represents differentiation with respect to conformal time ${\displaystyle \tau }$.

Taking ${\displaystyle B=E=0}$ (i.e. setting ${\displaystyle \mathrm{\Phi }\equiv -C}$ and ${\displaystyle \mathrm{\Psi }\equiv A}$), the Newtonian gauge is

${\displaystyle d{s}^2=a^2(\tau )\left[(1+2\mathrm{\Psi })d{\tau }^2-(1-2\mathrm{\Phi }){\delta }_{ij}d{x}^{i}d{x}^j\right]}$.

Note that in the absence of anisotropic stress, ${\displaystyle \mathrm{\Phi }=\mathrm{\Psi }}$.

A useful non-linear extension of this is provided by the non-relativistic gravitational fields.